Journal of Econometrics 167 (2012) 47–60
Contents lists available at SciVerse ScienceDirect
Journal of Econometrics
journal homepage: www.elsevier.com/locate/jeconom
The dynamics of US inflation: Can monetary policy explain the changes?
Fabio Canova a,∗ , Filippo Ferroni b
a
ICREA-UPF, CREI e CREMED, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain
b
Banque de France, 31 rue Croix des Petits Champs, 75001 Paris, France1
article
info
Article history:
Received 6 September 2006
Received in revised form
10 July 2010
Accepted 9 August 2011
Available online 9 November 2011
abstract
We investigate the relationship between monetary policy and inflation dynamics in the US using a
medium scale structural model. The specification is estimated with Bayesian techniques and fits the data
reasonably well. Policy shocks account for a part of the decline in inflation volatility; they have been less
effective in triggering inflation responses over time and qualitatively account for the rise and fall in the
level of inflation. A number of structural parameter variations contribute to these patterns.
© 2011 Elsevier B.V. All rights reserved.
JEL classification:
E52
E47
C53
Keywords:
New Keynesian model
Bayesian methods
Monetary policy
Inflation dynamics
1. Introduction
The US economy has gone through a number of important
structural changes over the last forty years. For example, the
level of inflation and of nominal interest rates shows an inverted
U-shaped pattern, rising at the end of the 1970s and falling at the
beginning of the 1980s; while the persistence and the volatility of
inflation have dramatically declined since the mid-1980s; see e.g.
Stock and Watson (2002). These patterns are well documented in
the literature. What is still to be determined is the cause of these
changes.
The prevailing view suggests that the run-up of inflation
occurred because monetary authorities believed that there was an
exploitable trade-off between inflation and output. Since output
was low following the oil shocks of the 1970s, the temptation
to inflate was strong. However, the option of keeping inflation
temporarily high was unfeasible: in the medium run, inflation
reached a higher level with output settling at its potential.
∗ Corresponding author. Tel.: +34 93 542 1926, +34 93 542 2668; fax: +34 93 542
1860, +34 93 542 2826.
E-mail addresses: [email protected], [email protected]
(F. Canova).
1 The views expressed in this paper do not necessarily reflect those of the Banque
de France.
0304-4076/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.jeconom.2011.08.008
Since the 1980s, central banks learned that the output–inflation
trade-off was not exploitable and concentrated on the objective
of fighting inflation. A low inflation regime ensued, and the
larger predictability of monetary policy made the macroeconomic
environment less volatile (see e.g. Sargent (1999), Clarida et al.
(2000), and Lubik and Schorfheide (2004)). There are two
alternative views as regards this prevailing wisdom: one focuses
on ‘‘real’’ causes (see e.g. McConnell and Perez Quiroz (2000),
Campbell and Herkovitz (2006)) and the other hinges on ‘‘good
luck’’ (see e.g. Bernanke and Mihov (1998), Leeper and Zha (2003),
Sims and Zha (2006)) to explain the changes in the level and in the
autocovariance function of inflation.
One reason for this heterogeneity of explanations is that the
empirical strategy used to study the issue matters. In general,
VAR based evidence tends to support the good luck hypothesis;
calibration exercises point to real reasons for the changes; and
structural econometric analyses favor the idea that monetary
policy is responsible for the observed variations (see, e.g., Ireland
(2001), Lubik and Schorfheide (2004) and Boivin and Giannoni
(2006)). However, while structural VAR exercises allow for time
varying coefficients and variances, the evidence produced by
more structural calibration or econometric analyses is mostly
restricted to arbitrarily chosen subsamples. Because inflation
and the nominal rate displayed an inverted U-shaped pattern,
subsample evidence may depend on the selected break point.
Fernandez Villaverde and Rubio Ramirez (2008) and Justiniano
48
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
and Primiceri (2008) have estimated evolving structural models
but their conclusions are only suggestive, because computational
complexities force them to consider variations only in a subset of
the parameters. Given that one expects important covariations in
the evolution of structural parameters, allowing only a subset of
the parameters to change may bias inference. Hence, it is of interest
to know whether less computationally intensive and yet intuitively
appealing structural methods can tell us more about the nature of
the changes experienced by US inflation.
This paper provides a step in that direction by estimating a
structural model over rolling samples of fixed length with Bayesian
techniques. Bayesian methods, which have become popular tools
for bringing DSGE models to the data thanks to the work of
Smets and Wouters (2003) and Del Negro and Schorfheide (2004)
among others, have inferential and computational advantages over
traditional limited and full information classical techniques when
dealing with models which are known to be a misspecified description of the data. In these situations, unrestricted classical estimates
are often unreasonable or on the boundary of the parameter space
and tricks must be employed to produce economically sensible estimates. Furthermore, asymptotic standard errors attached to classical estimates – which are constructed assuming that the model is
‘‘true’’ – are meaningless. Rolling samples allow us to use relatively
standard techniques to study the nature of the time variations
present in interesting parameters while maintaining some form of
rationality in the economy and keeping computational costs manageable. For example, in contrast to that of Fernandez Villaverde
and Rubio Ramirez (2008), our setup allows the use of Kalman filtering techniques in building the likelihood function and permits
time covariations in all the parameters.
The specification that we consider deviates somewhat from
what is standard in the literature by allowing money to play a
role. The stock of money has been neglected in all recent monetary
policy discussions (see e.g. Woodford (2003)) and Ireland (2004)
provided some empirical evidence supporting this approach. In our
setup real balances can potentially affect the Euler equation and
the growth rate of real balances is allowed to enter the monetary
policy rule. Since we will use loosely specified but proper priors
in the estimation, the data will decide whether these features are
important in characterizing the experience. Overall, the statistical
fit of the model looks satisfactory, in particular, in comparison
with other structural specifications. We estimate the preferred
specification a number of times over rolling samples, analyze
the time evolution of interesting inflation statistics, measure the
contribution of monetary policy to the observed changes and study
the evolution of the structural parameters.
Our model captures the fall in inflation volatility over time and
attributes part of the changes to monetary policy shocks. We detect
level but not shape differences in the transmission of policy shocks
which tend to make inflation less reactive to policy disturbances
as time goes by. Finally, variations in the level of inflation are
qualitatively related to policy shocks: had those been absent, the
rise of the 1970s and the fall of 1980s would have been much more
modest.
A number of structural changes drive these results. We find
support for the conjecture that the Fed had a much stronger
dislike for inflation but also notice that in the latest samples the
coefficient resembles the one obtained at the beginning of the
sample. Moreover, the estimate of the long run coefficient on
monetary aggregates has been steadily declining over time. We
detect, in agreement with the good luck hypothesis, variations in
the posterior mean estimate of the variance of the policy shocks.
Nevertheless, as in Sims and Zha (2006), the variations that we
discover are typically reversed over time. Finally we also find,
in consistency with non-monetary explanations of the facts, that
important private sector parameters such as the slope of the
Phillips curve and the variability of real demand shocks have
significantly changed in the later samples.
In sum, we find, in consistency with the conclusions of Gambetti
et al. (2008), that a combination of causes appears to be responsible
for the changes in the level and the autocovariance function of US
inflation over the last forty years: changes in the variance of the
shocks, in the parameters regulating private sector behavior and in
the policy rule all more or less contributed to explain why inflation
rose and fell, and why inflation volatility subsided.
The rest of the paper is organized as follows. Section 2 describes
the model, the estimation technique and the diagnostics used to
evaluate the quality of the model’s approximation to the data.
Section 3 presents estimation results for the full sample. Section 4
reports the time profile of inflation statistics over the rolling
samples. Section 5 interprets these time profiles in terms of rolling
structural parameter estimates. Section 6 concludes.
2. The framework of the analysis
2.1. The theoretical model
We consider a medium scale model featuring several shocks
and frictions. Households maximize a utility function which
depends on three arguments (money, consumption and leisure),
and money and consumption are potentially non-separable. Labor
is differentiated over households, so there is some monopoly over
wages. Households allocate wealth between cash and a riskless
bond, and bond demand is perturbed by a preference disturbance
(as in Smets and Wouters (2007)). Households also rent capital
services to firms and decide how much capital to accumulate. As
the rental price of capital goes up, the capital stock can be used
more intensively according to some schedule cost. There are two
kinds of firms: a final good representative firm that aggregates
intermediate goods, and a continuum of intermediate producing
firms that combine labor and capital in a monopolistic competitive
market where price decisions are subject to a Calvo lottery. Prices
that cannot be optimally adjusted are assumed to be partially
indexed to past inflation. Similarly, in the labor market unions
sell differentiated units of labor in a monopolistic competitive
environment with a Calvo type scheme. When unions receive
positive signals, they are allowed to re-optimize wages; otherwise
they adjust wages, indexing them to past inflation. Finally, profits
generated from the imperfectly competitive intermediate goods
and the labor markets are redistributed to households. The nominal
interest rate is controlled by a monetary authority who set it in
reaction to inflation, output gap and real balances.
The equations that we employ can be derived from first
principles—optimizing and forward looking consumers and firms
and general equilibrium considerations. Since derivations of this
type exist in the literature (see e.g. Smets and Wouters (2003)
and Smets and Wouters (2007)), we simply present the optimality
conditions and highlight how they link to the objective functions
and the constraints of the agents. The system in log-linear form is
ω1 θt = −ct + hct −1 − ω2 mt + ω2 et
(1)
ω3 lt = θt + w
(2)
HH
t
θt = −ω5 ct + hω5 ct −1 − ω4 mt + ω4 et −
1
R−1
(ϵtb + rt )
(3)
zt = a′ /a′′ rtk
(4)
kst
(5)
= zt + kt −1
kst = wt + lt − rtk
mct = (1 − α)wt + α
(6)
rtk
−ϵ
a
t
(7)
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
yt =
yt =
Ψ
Ψ
Ψ
ϵta +
α(wt − rtk ) +
lt
Ψ −1
Ψ −1
Ψ −1
C
K
ct +
Y
Y
(8)
δ it + ϵ g ϵtg + α zt
(9)
kt = (1 − δ)kt −1 + δ it + δϵti
(10)
rt = λr rt −1 + λy yt + (λp + λm )πt + λm (mt − mt −1 ) + ζtr
θt = Et [θt +1 + ϵtb + rt − πt +1 ]
(11)
qt = Et [β r k rtk+1 + β(1 − δ)qt +1 − θt + θt +1 ]
(12)
(1 + β)S it = Et [qt + ϵ + S it −1 + β S it +1 ]
 

ϵw
ϵtw + wt −1
(kw + βξw )wt = Et kw wtHH + w
ϵ +1

+ βwt +1 − (1 + iw β)πt + iw πt −1 + βπt +1
(13)
′′
πt = Et
i
t

ip
1 + β ip

+ kp
πt −1 +
′′
′′
β
1 + β ip
ϵp
mct + p
ϵp
ϵ +1 t
cc
(14)
p

(15)
m
− CUUmmc . Eq. (1) relates the marginal utility of consumption, θt , to
current consumption, ct , past consumption (controlled by a degree
of habit, h), real balances, mt , and a money demand shock, et .
When preferences are separable in real balances and consumption,
∂ 2 U (c ,m)
∂C∂m
w
w
, where 1 − ξw is the probability
with slope kw =
ξw
of re-optimizing wages. Eq. (15) is the New Keynesian Phillips
curve: current inflation depends positively on past and expected
inflation, and on marginal costs. This equation is perturbed by a
price markup disturbance, ϵt , and the slope is kp =
πt +1
L
where current consumption and money balances depend on their
expected values and the ex ante real interest rate, rt − Et πt +1 .
Eq. (12) is the Q equation that gives the value of capital stock,
qt , which depends positively on the expected future value of
the capital stock and on the real rental rate and negatively on
the marginal utility of consumption. Eq. (13) is an investment
equation: the current value of the investment depends on the past
and expected future values of the investment and on the current
value of the stock of capital. Eq. (14) gives the dynamics of the
real wage, which moves sluggishly because of the wage stickiness
and the partial indexation assumption; the wage responds to past
and future expected real wages, and to the (current, past and
expected) movements in inflation. The real wage depends also
on the wage paid by the union and on the wage markup, ϵtw ,
(1−ξ )(1−βξ )
where variables with the time subscript are log deviations from the
steady state, and variables without are variables at the steady state.
The ω’s are convolutions of steady state values and parameters
that capture the curvature of the utility function. In particular,
LV
mm
cm
, ω3 = − VLL , ω4 = − mU
and ω5 =
ω1 = − CUUc , ω2 = mU
CU
U
cc
49
= 0, so ω2 = 0 and ω5 = 0 and real balances do not
enter in the Euler equation. Unions pay a wage, wtHH , constant
across households (HH), which reflects the marginal rate of
substitution between working, lt , and consumption; see Eq. (2).
Eq. (3) is a money demand equation which depends on current and
past consumption, the nominal interest rate, rt , and a preference
disturbance, ϵtb . Current capital services, kst , are a function of the
capital installed in the previous period, kt −1 , and the degree of
capital utilization, zt (see Eq. (5)). Household cost minimization
implies that the degree of capital utilization is a positive function
of the rental rate of capital, as in Eq. (4). Eq. (6) relates the capital
per worker to the cost of capital, rtk , and the cost of labor, wt .
In Eq. (7) the marginal cost, mct , is determined by the real cost
of the two factors in production, rtk and wt , with weights given
by their respective share in production, net of the total factor
productivity disturbance, ϵta . Final output, yt , is produced using a
Cobb–Douglas production function (see Eq. (8)), where Ψ captures
the fixed cost in production, and α is the capital share. Total
g
output is absorbed by exogenous government spending, ϵt , by
investment, it , by consumption, ct , and by the capital utilization
cost, zt (see Eq. (9)). New installed capital is formed by the flows of
investment and the old capital net of depreciation, (1 − δ)kt −1 ;
see Eq. (10). The capital accumulation equation is hit by the
investment-specific technology disturbance ϵti . Eq. (10) represents
a policy rule. The specification is standard in the first three terms,
reflecting an interest rate smoothing desire and the wish to
respond to fluctuations in the output and inflation. We allow the
policy rule to depend on the growth rate of nominal balances in
order to mimic concerns that Central Banks had over monetary
aggregates for part of the sample and let λm ≥ 0. The disturbance
to this equation represents a monetary policy shock which is,
by construction, orthogonal to the other structural disturbances.
Eq. (11) describes the dynamics of consumption and real balances,
(1−ξp )(1−βξp )
,
1+ip β
where β is the time discount factor, and ξp is the probability of
keeping the prices fixed. The system is driven by eight exogenous
processes: et = ρe et −1 + ζte (money demand), ϵtb = ρb ϵtb−1 + ζtb
(preference), ϵta = ρa ϵta−1 + ζta (technology), ϵti = ρi ϵti −1 +
ζti (investment specific), ln 
ϵtg − ln ϵ g = ϵtg = ρg ϵtg−1 + ζtg
p
(government), ln 
ϵt − ln ϵ p = ϵtp = ρp ϵtp−1 + ζtp (price markup),
w
w
ln 
ϵt − ln ϵ = ϵtw = ρw ϵtw−1 + ζtw (wage markup) and ϵtr = ζtr
(monetary policy shock), where the ζ ’s are normal i.i.d. shocks.
We assume that the investigator observes output, consumption,
investment, hours worked, real wages, real balances, the inflation
rate, and the nominal interest rate.
The available sample goes from 1959:1 to 2006:1 and the data
set is obtained from the FRED database at the Federal Reserve Bank
of St. Louis. We stop at 2006 to maintain comparability with the
existing literature and this avoids headaches as regards how to deal
with the recent financial crisis episode. For real balances we use
real M2 deflated by the GDP deflator. The inflation rate is measured
by the growth rate of the GDP deflator, and the nominal interest
rate by the federal funds rate, and real variables are scaled by the
civilian noninstitutional population (CNP16OV) to transform them
to per capita terms. Despite these transformations, some series
still display upward trends. We therefore separately eliminate this
from their log using a simple quadratic specification. While the
resulting fluctuations display long periods of oscillation, they are
overall stationary as the model assumes (see Fig. 1).
As a referee has pointed out, one feasible alternative to the
strategy that we use to match the data to the model’s counterparts
is to allow the technology or the investment specific shock to be
non-stationary and remove the upward trend in output and real
balances using a model consistent method. We do not follow this
approach for three reasons. First, when these shocks have a unit
root, output and real balances share the same trend, which is not
the case with the available data. Second, the data set transformed
with a model based trend still displays a clear non-stationary
behavior and this makes parameters estimates unreliable and the
quality of the fit quite poor. Finally, it is unclear whether all noncyclical fluctuations can be safely attributed to non-stationary
technology shocks. For example, Chang et al. (2006) have recently
fit a model with non-stationary preference shocks to US data with
good results.
2.2. The prior and the estimation technique
The model (1)–(10) and (10)–(15) contains up to 38 parameters:
18 structural ones η1 = (h, π , β, a′ /a′′ , S ′′ , ξw , iw , ϵw , ξp , ip , ϵp ,
λr , λy , λm , λπ , ϵ g , δ, α), 5 semi-structural ones η2 = (ω1 , ω2 , ω3 ,
50
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
Fig. 1. Data used in the estimation.
ω4 , ω5 ), and 15 auxiliary ones, η3 = (ρe , ρb , ρa , ρi , ρg , ρw , ρp ,
σe , σb , σa , σi , σg , σr , σw , σp ). In our exercises we fix a few of them
(the time discount factor β = 0.995, the capital depreciation
rate δ = 0.025, the steady state government spending over GDP
ϵ g = 0.18, the capital share α = 0.3, and the steady state inflation
π = 1.006) since they are not identifiable from the data that
2. For each i = 1, . . . , k, where k is the dimension of the estimated
parameter vector, set ηi = ηi−1 with probability 1 − p and
ηi = ηi∗ with probability p, where ηi∗ = ηi−1 + vi and v =
[v1 , . . . , vk ] follows a multivariate normal distribution and p =
we use, and we obtain posterior distributions for the remaining
elements of η = (η1 , η2 , η3 ) using loosely specified but proper
priors.
We assume univariate prior densities for each of the parameters
even if, in principle, there should be some correlation structure
among the priors (see e.g. Del Negro and Schorfheide (2008)). Since
all the priors are proper, posterior distributions are proper and
various specifications can be compared with simple odd ratios.
The prior shapes that we use in our benchmark specification are
relatively standard (see Table 1), and very similar to the ones used
in Smets and Wouters (2007) but with much larger prior standard
deviations.
The model can be solved with standard methods. Its solution
has a state space format:
3. Repeat steps 1 and 2 L̄ + L times and keep the last L draws.
At the end of the routine one has L draws with which to conduct
the structural analysis.
Two important issues concern the convergence of simulated
draws, that is, what the size of L̄ is, and the acceptance rate. We
set the number of iterations to 500,000, checked for convergence
using the cumulative sum of the draws (CUMSUM) statistics and
found that convergence is achieved typically after 200,000 draws.
For inference we discard the first half of the chain and keep one
from every 250 draws, so that we remove the correlation among
draws and have 1000 draws from the posterior distribution to work
with. To get reasonable acceptance rates, it is important to properly
select the variance of vi . If the acceptance rate is ‘‘too small’’ the
chain will not visit the parameter space in a reasonable number of
iterations. If it is too high, the chain will not stay long enough in
the high probability regions. We set the variance of vi to target an
acceptance rate in the range 20%–40%.
The chosen prior shapes reflect the restrictions on the support
of the parameter space. Various parameterizations of the Beta
distributions were tried and results are, by and large, insensitive
to settings of the parameters controlling the location of these
distributions. Finally, the choice of prior distributions for the
policy parameters implies that multiple equilibria are unlikely to
characterize the data.
x1t +1 = A1 (η)x1t + A2 (η)ζt +1
(16)
x2t = A3 (η)x1t
(17)
g
where x2t = [mt , ht , yt , ct , it , wt , πt , rt ], ζt = [ζ , ζ , ζ , ζ , ζt ,
ζtr , ζtp , ζtw ] and the matrices Ai (η), i = 1, 2, 3, are complicated
nonlinear functions of the η’s.
Bayesian estimation of (16) and (17) is simple: given some η,
and a sample t , . . . , T , we compute the likelihood, denoted by
f (y[t ,T ] |η), by means of the Kalman filter and the prediction error
decomposition. Then, for any specification of the prior distribution,
denoted by g (η), the posterior distribution for the parameters
e
t
g (η)f (y
b
t
a
t
i
t
L(y[t ,T ] |η∗ )g (η∗ )
min{1, L(y |η i )g (ηi ) }.
i−1
[t ,T ] i−1
2.3. A measure of the fit and model comparisons
|η)
[t ,T ]
is g (η|y[t ,T ] ) =
. The analytical computation of the
f (y)
posterior is impossible in our setup since the denominator of the
expression, f (y), requires the integration of g (η)f (y[t ,T ] |η) with
respect to η, which is a high dimensional vector. In order to
obtain draws from the unknown posterior distribution we use the
following (Metropolis) algorithm:
1. Choose an η0 . Evaluate g (η0 ); use the Kalman filter to evaluate
the likelihood L(y[t ,T ] |η0 ).
To assess the quality of our model’s approximation to the data
we have estimated a number of interesting structural benchmarks.
We present the marginal likelihood (ML) of our model and of
four alternative specifications where e.g. real balances play no role
in the Euler equation, or real balances do not enter the policy
rule, or both. For each model Mj , we approximate L(y[t ,T ] |Mj ),
the marginal likelihood of Mj , using [ 1L
j

f (ηl )
−1
.
j
j
l L(y
[t ,T ] |ηl ,Mj )g (ηl |Mj )
]
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
51
Table 1
Prior and posterior statistics; marginal likelihoods (ML) and acceptance rates for the four specifications.
η
Prior
Posterior
Description
Habit in cons
−Uc /(C ∗ Ucc )
(m ∗ Ucm )/(Ucc C )
−(l ∗ Vll )/Vl
−(m ∗ Umm )/Um
Umc C /Um
k utiliz
Inv adj cost
Wage stickiness
Wage indexation
Elast variety labor
Price stickiness
Price indexation
Elast variety goods
MP autoregressive
MP resp to GDP
MP resp to money
MP resp to inflation
AR money demand
AR preference
AR technology
AR investment
AR government
AR wage markup
AR price markup
SD money demand
SD preference
SD technology
SD investment
SD government
SD monetary policy
SD wage markup
SD price markup
h
ω1
ω2
ω3
ω4
ω5
a′ /a′′
S ′′
ξw
iw
ϵw
ξp
ip
ϵp
λr
λy
λm
λπ
ρe
ρb
ρa
ρi
ρg
ρw
ρp
σe
σb
σa
σi
σg
σr
σw
σp
Density
beta(14, 6)
gam(4, 0.25)
gam(4, 0.25)
norm(2, 0.5)
gam(4, 0.25)
gam(4, 0.25)
beta(5.05, 5.05)
norm(1, 1)
beta(12, 12)
beta(1, 1)
norm(10, 2)
beta(12, 12)
beta(1, 1)
norm(10, 2)
beta(10, 2)
norm(0.15, 0.05)
norm(0.3, 1)
norm(2, 1)
beta(2.62, 2.62)
beta(2.6, 2.6)
beta(2.6, 2.6)
beta(2.6, 2.6)
beta(2.6, 2.6)
beta(2.6, 2.6)
beta(2.6, 2.6)
igam(3, 0.02)
igam(3, 0.02)
igam(3, 0.02)
igam(3, 0.02)
igam(3, 0.02)
igam(3, 0.02)
igam(3, 0.02)
igam(3, 0.02)
Mean (sd)
0.7(0.1)
1(0.5)
1(0.5)
2(0.5)
1(0.5)
1(0.5)
0.5(0.15)
1(1)
0.5(0.1)
0.5(0.3)
10(2)
0.5(0.1)
0.5(0.3)
10(2)
0.8(0.1)
0.15(0.05)
0.3(1)
2(1)
0.5(0.2)
0.5(0.2)
0.5(0.2)
0.5(0.2)
0.5(0.2)
0.5(0.2)
0.5(0.2)
0.01(0.01)
0.01(0.01)
0.01(0.01)
0.01(0.01)
0.01(0.01)
0.01(0.01)
0.01(0.01)
0.01(0.01)
p
0.1
0.3
0.5
0.7
0.9
Acceptance
f (η ) = p
−1
(2π )
−k/2
Rate


j
j
exp −1/2(ηl − η̄j )Σ j (ηl − η̄j )′
ηlj ,
Σ j = L l (η − η̄j )(η − η̄j )′ . Moreover, η is such that (η −
η̄j )Σ j (ηlj − η̄j )′ ≤ χp2 (kj ), where χp2 (kj ) is the p quantile of a
j
where ηl is draw l for the parameters η of model j, η̄j =

1
j
l
j
l
R1
R2
R3
Median (sd)
0.35(0.06)
0.87(0.11)
0.58(0.08)
0.54(0.07)
0.09(0.05)
0.37(0.14)
0.09(0.03)
3.74(0.606)
0.15(0.02)
0.46(0.25)
10(2.25)
0.83(0.01)
0.04(0.05)
10.8(1.76)
0.79(0.04)
0.13(0.01)
2.91(0.12)
3.48(0.44)
0.95(0.01)
0.83(0.03)
0.98(0.01)
0.71(0.05)
0.94(0.01)
0.96(0.01)
0.87(0.03)
0.015(0.002)
0.012(0.001)
0.009(0.000)
0.043(0.007)
0.034(0.002)
0.033(0.002)
0.007(0.001)
0.034(0.006)
0.09(0.02)
1.66(0.16)
–
0.38(0.04)
3.76(0.20)
–
0.07(0.03)
5.08(0.33)
0.04(0.01)
0.13(0.11)
9.9(1.9)
0.44(0.08)
0.08(0.09)
10.1(0.48)
0.8(0.04)
0.12(0.011)
–
3.42(0.62)
0.98(0.01)
0.84(0.03)
0.94(0.01)
0.24(0.03)
0.92(0.01)
0.96(0.01)
0.92(0.01)
0.020(0.002)
0.012(0.001)
0.009(0.000)
0.100(0.008)
0.034(0.002)
0.031(0.003)
0.007(0.000)
0.016(0.001)
0.37(0.01)
0.07(0.02)
1.46(0.05)
1.45(0.03)
0.12(0.01)
0.5(0.03)
0.13(0.01)
2.12(0.08)
0.34(0.01)
0.64(0.03)
12.9(0.87)
0.44(0.02)
0.08(0.01)
12.5(0.65)
0.92(0.02)
0.14(0.01)
–
2.04(0.01)
0.98(0.02)
0.83(0.02)
0.86(0.02)
0.70(0.02)
0.88(0.02)
0.92(0.002)
0.97(0.001)
0.012(0.001)
0.012(0.000)
0.009(0.001)
0.024(0.003)
0.033(0.001)
0.026(0.001)
0.008(0.001)
0.016(0.001)
0.2(0.04)
1.58(0.11)
–
0.16(0.03)
0.81(0.07)
–
0.35(0.06)
0.71(0.12)
0.21(0.01)
0.63(0.07)
11.6(0.76)
0.77(0.01)
0.07(0.05)
9.5(2.3)
0.8(0.04)
0.14(0.01)
2.9(0.31)
2.27(0.31)
0.97(0.01)
0.83(0.03)
0.80(0.03)
0.91(0.02)
0.90(0.02)
0.76(0.03)
0.78(0.024)
0.007(0.001)
0.012(0.001)
0.009(0.000)
0.007(0.001)
0.034(0.002)
0.031(0.003)
0.005(0.000)
0.030(0.003)
−541.64
−540.47
−539.88
−539.53
−539.08
−453.56
−452.46
−451.94
−451.59
−451.24
−120.03
−118.92
−118.39
−118.01
−117.57
33.93
49.33
40.55
ML
−113.89
−112.79
−112.25
−111.92
−104.92
32.05
Following Geweke (1998), we define for some p ∈ (0, 1)
j
l
UR
j
l
1
L

l
j
l
chi-square distribution where kj is the number of parameters in
model j.
To provide additional information on the success of the
estimation process, we also examine the properties of the
structural residuals and compare conditional responses to those
obtained from more restricted specifications. Both statistics can
tell us something about specification problems and therefore
nicely complement what marginal likelihood comparisons give us.
3. Full sample estimation and specification searches
We start estimating the model for the full sample,
1959:1–2006:1. We are interested in verifying that it fits the data
reasonably well and, therefore, can be used to undertake the type
of analysis that we care about; and in examining whether the
model’s parameterizations can be simplified and certain specification choices matter for the results. Table 1 presents the posterior
medians and standard deviations for (i) a specification which employs the full set of parameters (UR), (ii) a specification which employs a separable utility function, ω2 = ω5 = 0, and no reaction
of the interest rate to real balances, λm = 0 (R1 ), (iii) a specification which assumes no reaction of the interest rate to real balances,
λm = 0 (R2 ), and (iv) a specification which employs a separable
utility function, ω2 = ω5 = 0 (R3 ).
A few aspects of the table deserve comments. First, priors and
posteriors tend to have different locations, spreads and, in several
cases, shapes. Therefore, the sample appears to be informative
about the properties of many of the parameters. Second, the mean
estimates of ω2 and ω5 are positive and the posterior distribution
is tight, both in an absolute sense and relative to the prior. The
elasticities of the three standard arguments in the utility function,
i.e. ω1 , ω3 and ω4 , have the expected signs.
Third, the parameters controlling the backward looking components of the Euler equation and the Phillips curve are small (h
and ip )—smaller than those obtained, for example, by Smets and
Wouters (2003) and Justiniano and Primiceri (2008). One reason
for this is that the preliminary data transformation that we use is
different.
Fourth, the coefficients in the policy rule imply a relative
mild smoothing desire and a somewhat more aggressive response
to inflation and real balances. We experimented with several
versions of the monetary policy rule, where for instance the
interest rate was allowed to react to real balances (instead of
real balances growth) or where the reaction is lagged rather than
contemporaneous. The fits of these specifications were poor and, in
particular, the estimate of the smoothing coefficient of the Taylor
rule was close to zero, making monetary policy residuals highly
52
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
Fig. 2. Impulse responses. From top to bottom, TFP, investment, preference, money demand, government, monetary policy. Solid (blue) line: unrestricted model, dotted
(red) line: restricted model R1 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
serially autocorrelated. The rule that we use has two important
features, which turns out to be important in the estimation: it
makes the relationship between interest rate and the right hand
side variables dynamic; money growth enters the rule.
To gain insights into the plausibility of the estimates, it is
interesting to analyze the transmission mechanism of shocks and
the decomposition of the variance implied by the unrestricted
and the restricted models. Fig. 2 reports the responses of the
interest rate, inflation, real balances and GDP to impulses in
supply (technology, investment) and demand (preference, money
demand and government) shocks and monetary policy in the
unrestricted and the R1 specifications.2 The dynamics induced by
supply shocks are in line with those of Smets and Wouters (2007),
where for example following a positive TFP shock, output increases
while inflation and the interest rate drop. An investment shock
produces similar co-movements of the interest rate, the inflation
and GDP and induces a drop in money demand. These dynamics
are roughly similar in models where money matters and where
money does not matter. The transmission of a monetary policy
shock is however different. In the baseline model, a monetary
policy tightening (positive monetary policy shock) increases the
interest rate and makes the M2 supply drop, generating a liquidity
effect. As a result, the economy contracts, and output and inflation
decrease. All the responses display some inertia and the response
of the GDP is quite sluggish.
When money does not matter, the correlation between
interest rate and money conditional on monetary policy shocks is
positive, implying that contractionary monetary policy generates
a significant increase of the interest rate and of the money supply.
Moreover, while money demand shocks have a non-negligible
2 In the appendix, we report the impulse response for specifications R and R ,
2
3
see Fig. A.1.
impact on nominal and real variables in the unrestricted model, in
the restricted specification all these effects are forced to zero and
are captured by other shocks.
Table 2 presents the k-step-ahead forecast error for GDP
and inflation in terms of structural shocks. The table contains
several results. First, while in the unrestricted specification the
money demand shock contributes significantly to the volatility
of GDP and inflation at horizons from 2 to 10 years, in the
restricted specifications money demand shocks play a negligible
role. In particular, they do not contribute to explaining the GDP
forecast error at any horizons, and they explain a small portion
of the volatility of inflation as long as λm ̸= 0. Second, while
markup shocks have some role in explaining GDP volatility at
short horizons, their impact tends to disappear in the long run,
and supply shocks, i.e. technology shocks, appear to be the
predominant source of GDP fluctuations. In line with Smets and
Wouters (2007), in the most restricted specification R1 wage
markup shocks counterintuitively explain a large portion of the
volatility of GDP at business cycles horizons. In sum, the dynamics
described by the unrestricted and the restricted specifications
differ substantially, in particular as far as the transmission of
monetary policy shocks and the share of the variance of output and
inflation attributed to shocks are concerned. Given the focus of the
investigation, models where money is not allowed to matter seem
unable to capture important features of the data.
It is also instructive to inspect the time path of the estimated
residuals to check for an interesting pattern left unexplained in
the estimation. Fig. 3 plots the smoothed residuals obtained with
the unrestricted specification. It is clear that during the 1970s,
the US economy experienced a series of negative monetary policy
shocks, while at the beginning of the 1980s, monetary policy is
characterized by a series of positive shocks.
Thus, the path of policy shock implies that monetary policy
was accommodative for most of the 1970s and became much
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
53
Table 2
k-step-ahead forecast error variance decomposition for GDP and inflation.
Steps ahead
UR
8
R1
16
40
R2
R3
8
16
40
8
16
40
8
16
40
Money shock
Preference
Technology
Investment
Government
Monetary policy
Wage markup
Price markup
Forecast error variance of GDP
24.7
14.9
11.1
0.6
0.0
0.0
21.9
61.2
82.2
17.1
1.9
1.5
6.3
3.0
1.8
0.6
0.0
0.0
5.1
11.2
0.4
23.6
7.8
3.0
0.0
0.0
59.4
0.3
0.4
0.0
14.3
25.6
0.0
0.0
58.9
0.1
0.1
0.0
19.3
21.5
0.0
0.0
48.3
0.0
0.0
0.0
41.7
10.0
0.0
0.0
19.5
3.4
0.0
0.0
17.7
59.4
0.0
0.1
3.2
3.9
0.0
0.0
8.6
84.2
0.0
0.0
0.0
0.1
0.0
0.0
0.8
99.1
0.1
0.0
20.1
50.2
1.3
0.0
10.5
17.7
14.2
1.1
0.6
57.4
21.3
2.2
1.2
1.9
0.7
0.0
3.7
90.0
0.8
0.1
1.5
3.3
Money shock
Preference
Technology
Investment
Government
Monetary policy
Wage markup
Price markup
Forecast error variance of Inflation
2.3
15.2
11.0
2.8
0.4
0.2
22.2
0.5
7.8
51.7
66.4
16.6
1.3
6.0
3.3
6.2
0.4
0.5
0.2
10.1
32.4
13.3
1.1
28.2
0.0
0.7
59.7
0.3
0.3
0.0
14.4
24.6
0.0
0.1
59.2
0.1
0.1
0.0
19.4
21.1
0.0
0.0
48.9
0.0
0.0
0.0
40.6
10.5
0.0
7.4
18.4
3.1
0.0
0.0
16.6
54.5
0.0
0.5
3.3
3.9
0.0
0.0
8.8
83.5
0.0
0.0
0.0
0.1
0.0
0.0
0.9
99.0
18.0
0.0
15.5
48.3
0.6
0.0
11.4
6.1
46.4
0.0
1.5
50.5
0.5
0.1
0.5
0.5
72.5
0.0
1.5
24.0
0.1
0.0
0.6
1.3
Fig. 3. Residuals; unrestricted specification.
tighter later on. Interestingly, the model also predicts a sequence
of negative interest shocks just after 2001. The monetary
policy residuals in the restricted specifications are similar; see
Fig. 4. However, the implied paths display a more pronounced
autoregressive component and residuals tend to be asymmetric
and skewed towards negative values.
Table 1, which reports the marginal likelihood of each model,
confirms previous conclusions. For a wide range of p-values,
the unrestricted specification is superior to all other structural
alternatives that we considered. Thus, allowing real balances
to play a role and specifying a policy rule which depends on,
among other things, the growth rate of nominal balances provide
important advantages in terms of the model’s fit and properties
of the residuals. We hope that this superiority will also help in
providing a better economic interpretation of the evidence.
4. Inflation dynamics
There is substantial controversy in the literature regarding
the role that monetary policy had in shaping the dynamics of
US inflation over the last forty years. We can shed some light
on the issue, conditionally on the model, by analyzing the time
profile of the estimated autocovariance function of inflation, by
measuring the contribution of policy shocks to the changes, and by
interpreting the evolution of these statistics in terms of changes in
the parameters of the model.
Roberts (2006) and Cecchetti et al. (2007) have studied, by way
of stochastic simulations and/or standard sensitivity analysis, how
changes in the parameters of a small scale model where money
plays no role affect inflation variability and persistence. Both
consider a somewhat hybrid model, where the Euler equation is ad
hoc, a time varying inflation target enters the policy rule and the
policy shock is persistent. Roberts finds that changes in the policy
parameters can account for the fall in the variance of inflation
but not its persistence. Cecchetti et al. find that two parameters
are crucial: the degree of indexation in the Phillips curve and the
variance of the policy shocks. In particular, only a considerable fall
in the indexation parameter is capable of explaining the absolute
fall in persistence that researchers have documented. In addition,
when indexation is low, a fall in the variance of the policy shocks
may also decrease inflation persistence.
The analysis that we conduct has two main differences with
these papers. First, unlike these authors, we examine the contribution of monetary policy shocks to the evolution of the variability and persistence of inflation (this is done in much the
54
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
Fig. 4. Monetary policy residuals; different specification.
same spirit as the analysis of Gambetti et al. (2008) but using a structural model rather than a VAR) and examine how
the evolution of certain parameter estimates changed the contribution of monetary policy shocks. To do this, let xt = [x1t , x2t ]′ and
represent the solution of the model as xt +1 = A1 (η)xt + A2 (η)ζt +1 .
Then var(xt ) = Σx solves Σx = A1 (η)Σx A1 (η)′ + A2 (η)Σζ A2 (η)′
and ACF(1)(xt ) = A1 (η)Σx , where Σζ is the covariance matrix of
the shocks and ACF(1) the first autocovariance coefficient. Since
monetary shocks are uncorrelated with the other disturbances, the
contribution of monetary shocks to these statistics can be computed by replacing Σζ with a matrix which is zero everywhere
except in the diagonal position corresponding to the monetary
shocks. As estimates of η change over time, we can assess how
the contribution of monetary shocks to these statistics has changed
and attribute the variations to particular parameters.
Second, while one would be tempted to present a graph
showing how inflation dynamics change when η varies within
a reasonable range, keeping all the other parameters fixed at
some value (for example, those estimated in the first column
of Table 1), one should be aware that such a graph would be
meaningless in our context, since the correlation structure of
estimates implies that the effect of changes in some parameters
cannot be measured independently of the others. To address the
questions of interest we have instead estimated a version of the
model on many overlapping samples. We started from the sample
[1959:1, 1976:1] and repeated estimation moving the starting date
by two years, while keeping the size of the sample constant at
17 years. Keeping a fixed window size is necessary to eliminate
differences produced by the different precisions of the estimates.
The last subsample is [1989:1–2006:1], so we produce 16 posterior
distributions for the parameters. Since the final sample roughly
corresponds to Greenspan’s tenure, we can compare the estimated
stance of monetary policy in the 1990s, where inflation was low,
with that of the 1970s, where inflation was high.
Given the large number of parameters involved and the sample
length (69), we have decided to reduce the number of parameters
to estimate and focus on those which are of interest. Thus, in each
estimation window, in addition to keeping fixed the time discount
factor β = 0.995, the capital depreciation rate δ = 0.025, the
steady state government spending over GDP ϵ g = 0.18, the capital
share α = 0.3, and the steady state inflation π = 1.006, we set
the two elasticities in the labor and intermediate good markets
to 13.
4.1. The volatility and the persistence of inflation
It is well documented in the literature that the times series
properties of US inflation have changed over time (see e.g. Stock
and Watson (2002)). Is the model able to capture these facts?
What is the contribution of policy shocks to these changes?
Fig. 5 presents statistics recursively computed from the data, the
estimates of these statistics obtained in the model and the statistics
produced if shocks of only one type – technology shocks (TFP
and investment), monetary policy shocks or real demand shocks
(government spending and preference) – were present. Money
demand shocks are not reported here since their role is minimal.
We measure volatility in the model using the population
unconditional variance and persistence using the population firstorder autocorrelation coefficient obtained using the solution of the
model and posterior mean estimates of the parameters in each
sample.
The statistics in the data show important time variations: for
example, there is a large drop in volatility when the samples
exclude the early 1970s, while the fall persistence is evident only
if windows including the 1980 and the 1990s are considered.
Interestingly, after that, inflation persistence starts rising again.
While the model is able to track quite well the volatility dynamics,
it fails to capture the drop in persistence experienced in the sample
including the 1980s and 1990s. Notice, however, that the fall in
inflation persistence is neither smooth nor long-lived—the same
pattern is maintained if the windows change in size of if the sample
is split into less overlapping windows. Thus, changes in volatility
and persistence could be due to different causes.
Interestingly, monetary policy shocks seem to have a role in
shaping the dynamics of inflation volatility and appear to be
responsible in part for the decrease observed in the latest samples.
Moreover, they track pretty well the rise and fall of volatility over
time. However, other shocks appear also to be responsible for the
evolution of the autocovariance function of inflation. Technology
shocks, for example, affect inflation volatility and persistence
mainly at the beginning of the sample and their importance is
falling over time. Oil shocks which materialized at the end of the
1970s are captured here as negative supply shocks, and contribute
to the inflation volatility at the beginning of the sample. Real
demand shocks matter also for the dynamics of inflation volatility
and persistence. However, their role is quite modest.
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
55
Fig. 5. Estimated variance (left) and persistence (right) of inflation. Dotted line: data, dashed line: full model, solid line: model only with technology shocks (TFP and
investment), first row, monetary policy shocks, second row, and real demand shocks (preference and government), last row.
Fig. 6. Responses of output and inflation to a normalized positive monetary policy impulse.
4.2. The transmission of monetary policy shocks
It is interesting to examine how the variations observed in Fig. 5
decompose into variations in the transmission of policy shocks
over time (for a given variance of the shocks) and variations in
the volatility of policy shocks. Gambetti et al. (2008) have shown,
in the context of a SVAR with time varying coefficients, that
there are variations in the size and the shape of the responses to
monetary shocks and in the variances of these shocks. What does
our structural analysis tell us about this issue? Fig. 6 presents the
responses to a normalized monetary policy shock in the 16 samples
that we have considered. Hence, the variations that it displays only
reflect changes in the structural parameters of the model and not
changes in the variance of the policy shocks (these will be analyzed
later on).
Overall, the shape of the responses to policy shock has not
changed much over time: when monetary policy is tight, inflation
and output fall. Quantitatively, the impact effect of monetary
shocks is somewhat reduced over time.
One other feature of the responses deserves some commentary:
the largest inflation response is always instantaneous. This may
appear surprising in view of the conventional VAR wisdom.
One should stress that much of the conventional wisdom is
derived using restrictions which are inconsistent with the theory
embedded in models like the one that we use and that when the
restrictions that the theory imposes are used, no strong delayed
response typically appears (see Canova and De Nicolo (2002)).
4.3. The rise and fall in the level of inflation
One crucial event that we would like our model to explain is
the rise in the level of inflation in the 1970s and the subsequent
fall in the 1980s. The conventional wisdom attributes these ups
and downs to a changing monetary stance, which was lax in the
1970s and became tight in the 1980s. The analysis of the previous
subsections is inconclusive since the statistics that we report are
not designed to shed light on this issue. Here we present the
results of a historical decomposition exercise where we take mean
posterior estimates of the parameters in four samples (1961–1978,
1965–1982, 1967–1984, 1969–1986), project the implied path
of inflation out-of-sample, and ask whether and by how much
policy shocks account for the deviation between the actual and
56
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
Fig. 7. The rise and fall of inflation.
the projected path. Formally, we decompose inflation in terms of
realized monetary policy shocks,

π̃T = SA3 (η̄T )
T


[A1 (η̄T )]T −m A2 (η̄T )ζ̂mĎ
(18)
m=1
where S is a selection matrix that picks inflation out of the vector
of observable variables, and η̄T is the mean posterior estimate
using the information up to T . {ζ̂mr }Tm=1 is the set of realized
Ď
monetary policy shocks in the sample considered and ζ̂m =
[0, 0, 0, 0, 0, ζ̂mr , 0, 0]. Then, we project inflation out-of-sample,
conditionally on time T information,
π̃Te+k = ET (π̃T +k ) = SA3 (η̄T ) [A1 (η̄T )]k


T

T −m
Ď
[A1 (η̄T )]
×
A2 (η̄T )ζ̃m .
(19)
m=1
Fig. 7, which presents the actual (de-meaned) and the
counterfactual inflation paths, displays interesting aspects. The
sequence of policy shocks that materialized after 1978:1 would
have implied an increase in the level of inflation, given the
parameter estimates we obtain. Hence, monetary policy was rather
accommodative in these periods. This pattern seems to change
after 1982:1. The sequence of policy shocks that materialized after
that date would have implied a considerable decline in the level of
inflation, given the parameter estimates. Hence, monetary policy
had become quite restrictive after this date and, if only policy
shocks had been present, inflation would have been very low
during the following three years, where information up to 1984:1
is used. Thus, policy shocks quantitatively predict movements in
the actual level of inflation, and can account for the direction of
the changes over the periods that we consider.
How does one reconcile the evidence of Figs. 5 and 7? Fig. 5
was concerned with the in-sample explanatory power of policy
shocks for volatility and persistence. Fig. 7 is instead concerned
with the out-of-sample explanatory power of these shocks for
the level of inflation in the next three years. Therefore, the fact
that these shocks may have some room for explaining changes
in the autocovariance function of inflation does not imply that
policy shocks are important for explaining changes in the level of
inflation.
4.4. Bringing the shocks of the 1970s into the Greenspan era
What would have happened to inflation if we could bring the
shocks prevailing in the 1970s into the ‘Great Moderation’ era?
Would we still observe low inflation with small volatility? Typical
Fig. 8. Conditional and actual paths of inflation.
counterfactuals are performed by moving the monetary policy
rule of Alan Greenspan back to the 1970s, which is equivalent to
asking whether monetary policy would have been able to stabilize
the economy given the shocks and the economic structure of the
1970s. Here, we take a different stand. We would like to know
whether the private sector and monetary policy would be able to
absorb and mitigate the realized shocks of the 1970s. The exercise
that we conduct, apart from being informative about the good
luck–good policy debate, is also less open to the Lucas critique
since we are manipulating only the shocks and not the structural
coefficients.
We compute the realized innovations of the sample that goes
−1 70
70
from 1967q1 to 1984q1 using 
ζt70
+1 = A2 (η )(xt +1 − A1 (η )xt ),
70
where η is the mean of the posterior distribution of the sample
[1967q1–1984q1]. We then simulate inflation assuming that the
economy evolves according to the estimated structure during the
1990s but it is fed with shocks different than those observed in
the 1990s. For example, when the shocks of the 1970s are fed in,
90
xst+1 = A1 (η90 )xst + A2 (η90 )
ζt70
is the mean of the
+1 where η
posterior distribution of the sample [1987q1–2006q1].
Fig. 8 plots the counterfactual paths that we generate and
the actual path of inflation during the ‘Great Moderation’ period.
The figure indicates that the counterfactual path for inflation
would have displayed more volatile and slightly more persistent
fluctuations than the actual ones of the 1990s. Thus, the intensity
of the shocks matters quite a lot.
4.5. Summary
The analysis of this section has shown that the model captures
the time profile of inflation volatility quite well and attributes
parts of the changes to monetary policy shocks. Nevertheless,
policy shocks do not have a predominant role in the decline: in
fact, the transmission of policy disturbances is broadly unchanged
over time. Hence, changes in the structural parameters somewhat
average out and do not amplify changes in the variance of the
monetary shocks. On the other hand, the rise and fall in the level
of inflation appear to be linked to the sign and, to some extent, the
magnitude of realized policy shocks. Without these shocks, the ups
and downs of inflation would have been much smaller. Finally, the
‘Great Moderation’ episode seems to be more likely linked with
the intensity of the shocks rather then changes in the monetary
policy behavior. Overall, the evidence suggests that changes in the
transmission of shocks other than policy disturbances could be as
or more important in explaining the changes in the autocovariance
function of inflation. To understand what factors may have given
these shocks an important role, we next turn to examining the
parameter estimates that we obtain in different samples.
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
57
Fig. 9. Rolling parameter median estimates.
5. Rolling parameter estimates
Fig. 9 presents the evolution of the posterior mean of interesting
(functions of the) parameters over different samples. For the sake
of readability, we omit posterior credible sets—since they are
relatively tight, the variations that we present are typically a
posteriori significant.
Consistent with the analysis of Clarida et al. (2000), the figure
shows that the short run policy coefficient for inflation increases
in the samples which mainly include the 1980s. Interestingly,
and consistently with the structural VAR evidence of Canova and
Gambetti (2009), the posterior mean estimate falls somewhat in
the latest samples. Hence, while Greenspan was tougher than
Burns in fighting inflation, the posterior distributions which
include only the 1990s and the beginning of the 2000s show
a reduced short run concern for inflation. This contrasts with
the dynamics displayed by the long run coefficient for nominal
variables (computed as (1 − λr )−1 (λπ + λm )). Here a clear
decreasing pattern emerges with a peak located at the beginning
of the sample. Note that the coefficient for the output (not
shown here) is stable over time and shows no sign of increase in
correspondence to the drop in the long run coefficient of nominal
aggregates.
The posterior mean of the standard deviation of the policy shock
(σr ) shows some variations over time. As convectional wisdom
suggests, there is an increase (to 0.035 from 0.015) in the samples
from (1959–1976) to (1967–1984) but this decrease is reversed
in the next few samples. This evidence is a bit surprising, but it
is in line with the evidence produced by the Markov switching
approach of Sims and Zha (2006), and the recursive analysis of
Gambetti et al. (2008). It is worth mentioning that decreasing the
length of the windows does not change this pattern: the increase
present in the last few years of the sample is consistent with the
more active role that the Fed has taken since 2001.
There are changes also in the posterior mean of other important
parameters. For example, there are shifts in responsiveness of the
output to real balances (see the real balance trade-off, ω2 /ω1 ) and
in the responsiveness of inflation to the marginal cost (Phillips
curve trade-off, κp ). The fact that the real balance trade-off is
increasing, coupled with a volatile but overall constant Phillips
curve trade-off, indicates that real balances may have played
different roles at different points in time. For example, it is possible
that real balances behaved as close proxies of consumer purchasing
power in the early samples while they proxy for segmented
asset markets in later samples (see e.g. Alvarez and Lippi (2009)).
Note that the price indexation mechanism ιp does not show any
significant falls as we move through the sample.
Interestingly, many other standard deviations display significant time variations. For instance, the standard deviations of
government spending, preference and money demand shocks increased at the beginning of the samples, and then steadily decreased, giving support to the idea that many structural shocks
have been less intense in the latter part of the sample; see e.g. Gali
and Gambetti (2009).
In sum, the parameter changes that we describe can explain
why shocks other than monetary policy disturbances have an
important role in explaining the variations in the volatility of
inflation documented in Fig. 5. It is because output changes in
response to these shocks are larger, and because changes in output
are proportionally more important for explaining inflation changes
(and because the size of these shocks fell) that we see a large
decline in the volatility of inflation over time.
6. Conclusions
This paper examines the contribution of policy shocks to the
dynamics of inflation using a medium scale structural model
estimated with US post-WWII data and Bayesian techniques over
rolling samples. The model belongs to the class of New Keynesian
structures that have been extensively used in the literature but
explicitly allows money to play a role. Bayesian techniques are
preferable to standard likelihood methods or to indirect inference
(impulse response matching) exercises, because the model that
we consider is a false description of the DGP of the data and
misspecification may be important. We show that our approach
delivers interesting estimates of the structural parameters when
priors are broadly non-informative and the policy reaction function
appropriately chosen. We also demonstrate that the model fits the
data reasonably well and that alternative structural specifications
produce lower marginal likelihoods and fail to capture important
aspects of the data.
Our model captures the fall in inflation volatility and attributes
parts of the changes to monetary policy shocks. We detect level but
not shape differences in the transmission of policy shocks which
tend to make inflation and output somewhat less reactive to policy
disturbances as time goes by. Finally, variations in the level of
58
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
inflation are qualitatively related to policy shocks: had those been
absent, the rise of the 1970s and the fall of the 1980s would have
been much more modest.
A number of structural changes drive these results. We find
support for the conjecture that the Fed had a much stronger
short run ‘‘dislike’’ for inflation in later samples but also notice
that the long run coefficient on nominal variables has been
steadily decreasing. We detect, in agreement with the good luck
hypothesis, variations in the posterior mean estimate of the
variance of the policy shocks. Perhaps surprisingly, we find that
these variations do not change the way in which monetary shocks
are transmitted to the economy.
In sum, several causes in combination are responsible for the
changes in the level and the autocovariance function of US inflation
over the last forty years: changes in the variance of the shocks,
in the parameters regulating the private sector and in the policy
rule all contributed to a greater or lesser extent to explaining why
inflation rose and fell and why inflation variability subsided. These
conclusions are rather different than those present in the literature,
with the exception of Gambetti et al. (2008).
There are a number of ways in which our analysis can be
extended. For example, the estimation approach that we employ
treats expectations as latent variables. However, measures of
output and inflation expectations do exist in the literature. While
these proxies are probably contaminated with measurement error,
it would be interesting to see whether they provide additional
or contrasting information about the issues at stake. Similarly,
it is important to consider additional statistics to the evaluation
process: while the model seems by and large well specified, it may
not capture the time profile of the dynamics of a particular variable
well. Finally, the use of alternative rolling estimation techniques,
such as those employed by Kapetanios and Yates (2008), can help
us to understand whether the conclusions are also robust along this
dimension. We leave all these extensions to future research.
Acknowledgments
We would like to thank the editor of this journal, two
anonymous referees, and the participants of seminars at the
University of Southampton, the University of Bern, the Bank of
England and the Swiss National Bank for comments and Evi Pappa
for constructive suggestions on an earlier draft of the paper. The
financial support of the Spanish Ministry of Education through the
grant SEJ-2004-21682-E and that of the Barcelona Graduate School
of Economics are gratefully acknowledged.
Appendix
The Smets and Wouter model with a non-separable utility
function.
In this section, we derive the equations of the model that differ
from those of the model of Smets and Wouters (2007), henceforth
SW. Households have a (non-separable in money) utility function
of the form

U
Ct − hCt −1 ,
Mt
et Pt
, Lt


= U Ct − hCt −1 ,

Mt
et Pt
− V (Lt ).
Let mt = Mt /Pt . The budget constraint in real terms is
m t + C t + It +
+
WtHH
Pt
Bt
ϵtb Rt Pt
Lt +
Rkt
Pt
+ T t = m t −1
1
πt
+
Zt Kt −1 − a(Zt )Kt −1 +
Pt −1 πt
Pt
+

Kt = (1 − δ)Kt −1 + ϵti 1 − S


It
It − 1
It .
The production side and the wage setting are identical to those
of SW. Also, the government budget constraint is similar with the
only exception being money, i.e.
Gt +
Bt
1
Pt −1 πt
+ mt −1
1
πt
= Tt +
Bt
ϵtb Rt Pt
+ mt .
The feasibility constraint is obtained simply by integrating across
households and adding the government budget constraint, i.e.
Gt + Ct + It + a(Zt )Kt −1 = Yt .
For the steady state, SW assume that S (1) = 0, S ′ (1) = 0, Z = 1
and a(Z ) = a(1) = 0. The steady state values are R = 1/βπ ,
1
α α (1−α)1−α mc 1−α
1
r k = 1/β − 1 + δ , r k = a′ (1), mc = 1+ϵ
,
p,w =
(r k )α
α
1−α
α
1−α
.
L/Y = w , K /Y = 1/β−1+δ , Ψ = (K /Y ) (L/Y )
From the first-order condition for consumption optimization
we obtain that


Uc
Ct − hCt −1 ,
Mt
et Pt


= θt .
We rewrite the right hand side of this equation as
t ) = Uc (t )
Uc ( 
Ct , m
t =
where 
Ct = Ct − hCt −1 and m
derivative are
mt
et
and mt =
Mt
Pt
. The partial
∂ Uc (t ) ∂ 
Ct
∂ Uc (t )
∂ Uc (t )
=
=

∂ Ct
∂
C
∂ Ct
∂
Ct
t

∂ Uc (t )
∂ Uc (t ) ∂ Ct
∂ Uc (t )
=
=−
h

∂ C t −1
∂
C
∂ Ct
∂
Ct
t −1
t
∂ Uc (t )
∂ Uc (t ) ∂ m
∂ Uc (t ) 1
=
=
 t ∂ mt
 t et
∂ mt
∂m
∂m
t
∂ Uc (t ) ∂ m
∂ Uc (t ) mt
∂ Uc (t )
=−
=−
.
 t ∂ et
t e2t
∂ et
∂m
∂m
At the steady state (e = 1), the partial derivatives are
∂ Uc
∂ Uc
=
= Ucc
∂ Ct
∂
Ct
∂ Uc
∂ Uc
=−
h = −hUcc
∂ C t −1
∂
Ct
∂ Uc
∂ Uc 1
=
= Ucm
t e
∂ mt
∂m
∂ Uc
∂ Uc m
=−
= −Ucm m.
t e2
∂ et
∂m
The log-linearized version of the condition is
t − Ucm m
Ucc C 
ct − hUcc C 
ct −1 + Ucm mm
et = θ
θt .
Since from the steady state equation we have that θ = Uc , the
latter can be rewritten as
Bt −1 1
Div P
W HH , and differentiate HH labor and sell it to labor packers in a
monopolistic competitive environment (profits are redistributed
to HH in terms of dividends). Labor packers then sell ‘‘packed’’ labor
to intermediate sector firms in a perfectly competitive market. The
capital accumulation law of motion is
Div L
Pt
where Div P and Div L are dividends paid by the intermediate sector
firms and labor unions. As in SW, labor unions pay a fixed wage,
t + ω2
ω1
θt = −
ct + h
ct −1 − ω2 m
et
Uc
cm
where ω1 = − CU
and ω2 = mU
.
CUcc
cc
From the optimality condition with respect to Lt , we obtain
VL (Lt ) = −θt WtHH /Pt = −θt wtHH
F. Canova, F. Ferroni / Journal of Econometrics 167 (2012) 47–60
59
Fig. A.1. Responses for the models R2 and R3 . From the top to bottom, TFP, investment, preference, money demand, government, monetary policy.
where WtHH is the wage paid by the unions. In the steady state,
VL = −θ W
HH

/P .
VLL L
lt = −θ w HH
θt − θ wHH w
tHH
where ω3 = − VLL .
L
Finally, maximization with respect to mt = Mt /Pt and bt =
Bt /Pt leads to
V L
Um
θt
Ct − hCt −1 ,

et Pt

1
ϵtb Rt
= β Et θt +1
1

1
= θt − β Et θt +1
πt +1

πt +1
and combining the two we obtain

Um
Ct − hCt −1 ,

Mt
et Pt
1
R
Um
Um

= Um
θt +

ϵ bt
rt +
R−1
R−1
ω3lt = 
θt + w
tHH
Mt

R−1
R−1


θt + θ
rt + θ

ϵ bt
R(R − 1)
R(R − 1)
t − Umm m
Umc C 
ct − hUmc C 
ct −1 + Umm mm
et
=θ 1−
Thus, the log-linear version of this optimality condition is

t − Umm m
Umc C 
ct − hUmc C 
ct −1 + Umm mm
et

1
= θt 1 −
Rt ϵtb

.

and dividing by Um we get
Umc C
Umm m
Umm m

t −

ct −1 +
m
et
Um
Um
Um
1
1

=
θt +
rt +

ϵ bt − ω5
ct + hω5
ct −1
R−1
R−1
1
1
t + ω4

− ω4 m
et = 
θt +
rt +

ϵ bt
R−1
R−1
Um

ct − h
Umc C
mc
mm
where ω5 = − CU
and ω4 = − mU
.
Um
Um
The remaining equations are the same as in SW.
References
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Um
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P


=θ 1−
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R

.
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t − Umm m
Umc C 
ct − hUmc C 
ct −1 + Umm mm
et

=θ 1−
1
R

1
1 b

θt + θ 2 R
rt + θ 
ϵt
R
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